Theorem mblfinlem4 37667

Description: Backward direction of ismblfin 37668. (Contributed by Brendan Leahy, 28-Mar-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)

Assertion

Ref	Expression
mblfinlem4	⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))

Distinct variable group:   𝑦,𝑏,𝐴

Proof of Theorem mblfinlem4

Dummy variables 𝑓 𝑔 𝑠 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ltso 11341	. . . 4 ⊢ < Or ℝ
2	1	a1i 11	. . 3 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → < Or ℝ)
3		simplr 769	. . 3 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (vol*‘𝐴) ∈ ℝ)
4		vex 3484	. . . . . . 7 ⊢ 𝑢 ∈ V
5		eqeq1 2741	. . . . . . . . 9 ⊢ (𝑦 = 𝑢 → (𝑦 = (vol‘𝑏) ↔ 𝑢 = (vol‘𝑏)))
6	5	anbi2d 630	. . . . . . . 8 ⊢ (𝑦 = 𝑢 → ((𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑢 = (vol‘𝑏))))
7	6	rexbidv 3179	. . . . . . 7 ⊢ (𝑦 = 𝑢 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑢 = (vol‘𝑏))))
8	4, 7	elab 3679	. . . . . 6 ⊢ (𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑢 = (vol‘𝑏)))
9		simprl 771	. . . . . . . . 9 ⊢ ((𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐴 ∧ 𝑢 = (vol‘𝑏))) → 𝑏 ⊆ 𝐴)
10		ovolss 25520	. . . . . . . . . . 11 ⊢ ((𝑏 ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ) → (vol*‘𝑏) ≤ (vol*‘𝐴))
11		sstr 3992	. . . . . . . . . . . . 13 ⊢ ((𝑏 ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ) → 𝑏 ⊆ ℝ)
12		ovolcl 25513	. . . . . . . . . . . . 13 ⊢ (𝑏 ⊆ ℝ → (vol*‘𝑏) ∈ ℝ*)
13	11, 12	syl 17	. . . . . . . . . . . 12 ⊢ ((𝑏 ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ) → (vol*‘𝑏) ∈ ℝ*)
14		ovolcl 25513	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ ℝ → (vol*‘𝐴) ∈ ℝ*)
15	14	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑏 ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ) → (vol*‘𝐴) ∈ ℝ*)
16		xrlenlt 11326	. . . . . . . . . . . 12 ⊢ (((vol*‘𝑏) ∈ ℝ* ∧ (vol*‘𝐴) ∈ ℝ*) → ((vol*‘𝑏) ≤ (vol*‘𝐴) ↔ ¬ (vol*‘𝐴) < (vol*‘𝑏)))
17	13, 15, 16	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑏 ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ) → ((vol*‘𝑏) ≤ (vol*‘𝐴) ↔ ¬ (vol*‘𝐴) < (vol*‘𝑏)))
18	10, 17	mpbid 232	. . . . . . . . . 10 ⊢ ((𝑏 ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ) → ¬ (vol*‘𝐴) < (vol*‘𝑏))
19	18	ancoms 458	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑏 ⊆ 𝐴) → ¬ (vol*‘𝐴) < (vol*‘𝑏))
20	9, 19	sylan2 593	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐴 ∧ 𝑢 = (vol‘𝑏)))) → ¬ (vol*‘𝐴) < (vol*‘𝑏))
21		simprrr 782	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐴 ∧ 𝑢 = (vol‘𝑏)))) → 𝑢 = (vol‘𝑏))
22		uniretop 24783	. . . . . . . . . . . . . . 15 ⊢ ℝ = ∪ (topGen‘ran (,))
23	22	cldss 23037	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ⊆ ℝ)
24		dfss4 4269	. . . . . . . . . . . . . 14 ⊢ (𝑏 ⊆ ℝ ↔ (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
25	23, 24	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) = 𝑏)
26		rembl 25575	. . . . . . . . . . . . . 14 ⊢ ℝ ∈ dom vol
27	22	cldopn 23039	. . . . . . . . . . . . . . 15 ⊢ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ (topGen‘ran (,)))
28		opnmbl 25637	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ∖ 𝑏) ∈ (topGen‘ran (,)) → (ℝ ∖ 𝑏) ∈ dom vol)
29	27, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ 𝑏) ∈ dom vol)
30		difmbl 25578	. . . . . . . . . . . . . 14 ⊢ ((ℝ ∈ dom vol ∧ (ℝ ∖ 𝑏) ∈ dom vol) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
31	26, 29, 30	sylancr 587	. . . . . . . . . . . . 13 ⊢ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (ℝ ∖ (ℝ ∖ 𝑏)) ∈ dom vol)
32	25, 31	eqeltrrd 2842	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → 𝑏 ∈ dom vol)
33		mblvol 25565	. . . . . . . . . . . 12 ⊢ (𝑏 ∈ dom vol → (vol‘𝑏) = (vol*‘𝑏))
34	32, 33	syl 17	. . . . . . . . . . 11 ⊢ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘𝑏) = (vol*‘𝑏))
35	34	ad2antrl 728	. . . . . . . . . 10 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐴 ∧ 𝑢 = (vol‘𝑏)))) → (vol‘𝑏) = (vol*‘𝑏))
36	21, 35	eqtrd 2777	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐴 ∧ 𝑢 = (vol‘𝑏)))) → 𝑢 = (vol*‘𝑏))
37	36	breq2d 5155	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐴 ∧ 𝑢 = (vol‘𝑏)))) → ((vol*‘𝐴) < 𝑢 ↔ (vol*‘𝐴) < (vol*‘𝑏)))
38	20, 37	mtbird 325	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (𝑏 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑏 ⊆ 𝐴 ∧ 𝑢 = (vol‘𝑏)))) → ¬ (vol*‘𝐴) < 𝑢)
39	38	rexlimdvaa 3156	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑢 = (vol‘𝑏)) → ¬ (vol*‘𝐴) < 𝑢))
40	8, 39	biimtrid 242	. . . . 5 ⊢ (𝐴 ⊆ ℝ → (𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} → ¬ (vol*‘𝐴) < 𝑢))
41	40	ad2antrr 726	. . . 4 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))} → ¬ (vol*‘𝐴) < 𝑢))
42	41	imp 406	. . 3 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ 𝑢 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}) → ¬ (vol*‘𝐴) < 𝑢)
43		1rp 13038	. . . . . . . . 9 ⊢ 1 ∈ ℝ+
44		eqid 2737	. . . . . . . . . 10 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑓)) = seq1( + , ((abs ∘ − ) ∘ 𝑓))
45	44	ovolgelb 25515	. . . . . . . . 9 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ ∧ 1 ∈ ℝ+) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)))
46	43, 45	mp3an3 1452	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)))
47		elmapi 8889	. . . . . . . . . . 11 ⊢ (𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
48		ssid 4006	. . . . . . . . . . . . . . 15 ⊢ ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ((,) ∘ 𝑓)
49	44	ovollb 25514	. . . . . . . . . . . . . . 15 ⊢ ((𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
50	48, 49	mpan2 691	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
51	50	adantl 481	. . . . . . . . . . . . 13 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ))
52		eqid 2737	. . . . . . . . . . . . . . . 16 ⊢ ((abs ∘ − ) ∘ 𝑓) = ((abs ∘ − ) ∘ 𝑓)
53	52, 44	ovolsf 25507	. . . . . . . . . . . . . . 15 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞))
54		frn 6743	. . . . . . . . . . . . . . . 16 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ (0[,)+∞))
55		icossxr 13472	. . . . . . . . . . . . . . . 16 ⊢ (0[,)+∞) ⊆ ℝ*
56	54, 55	sstrdi 3996	. . . . . . . . . . . . . . 15 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑓)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ*)
57		supxrcl 13357	. . . . . . . . . . . . . . 15 ⊢ (ran seq1( + , ((abs ∘ − ) ∘ 𝑓)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
58	53, 56, 57	3syl 18	. . . . . . . . . . . . . 14 ⊢ (𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
59		peano2re 11434	. . . . . . . . . . . . . . 15 ⊢ ((vol*‘𝐴) ∈ ℝ → ((vol*‘𝐴) + 1) ∈ ℝ)
60	59	rexrd 11311	. . . . . . . . . . . . . 14 ⊢ ((vol*‘𝐴) ∈ ℝ → ((vol*‘𝐴) + 1) ∈ ℝ*)
61		rncoss 5986	. . . . . . . . . . . . . . . . . 18 ⊢ ran ((,) ∘ 𝑓) ⊆ ran (,)
62	61	unissi 4916	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ran ((,) ∘ 𝑓) ⊆ ∪ ran (,)
63		unirnioo 13489	. . . . . . . . . . . . . . . . 17 ⊢ ℝ = ∪ ran (,)
64	62, 63	sseqtrri 4033	. . . . . . . . . . . . . . . 16 ⊢ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ
65		ovolcl 25513	. . . . . . . . . . . . . . . 16 ⊢ (∪ ran ((,) ∘ 𝑓) ⊆ ℝ → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ*)
66	64, 65	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ*
67		xrletr 13200	. . . . . . . . . . . . . . 15 ⊢ (((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ*) → (((vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
68	66, 67	mp3an1 1450	. . . . . . . . . . . . . 14 ⊢ ((sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ*) → (((vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
69	58, 60, 68	syl2anr 597	. . . . . . . . . . . . 13 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (((vol*‘∪ ran ((,) ∘ 𝑓)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
70	51, 69	mpand 695	. . . . . . . . . . . 12 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
71	70	adantll 714	. . . . . . . . . . 11 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑓:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
72	47, 71	sylan2 593	. . . . . . . . . 10 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1) → (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
73	72	anim2d 612	. . . . . . . . 9 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)) → (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))))
74	73	reximdva 3168	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑓)), ℝ*, < ) ≤ ((vol*‘𝐴) + 1)) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))))
75	46, 74	mpd 15	. . . . . . 7 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
76		rexex 3076	. . . . . . 7 ⊢ (∃𝑓 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → ∃𝑓(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
77	75, 76	syl 17	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → ∃𝑓(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
78	77	ad2antrr 726	. . . . 5 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ∃𝑓(𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
79		difss 4136	. . . . . . . . . . . . . 14 ⊢ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓)
80	79, 64	sstri 3993	. . . . . . . . . . . . 13 ⊢ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ
81		ovolcl 25513	. . . . . . . . . . . . 13 ⊢ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ*)
82	80, 81	ax-mp 5	. . . . . . . . . . . 12 ⊢ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ*
83	59, 82	jctil 519	. . . . . . . . . . 11 ⊢ ((vol*‘𝐴) ∈ ℝ → ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ))
84	83	ad4antlr 733	. . . . . . . . . 10 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ))
85		ovolss 25520	. . . . . . . . . . . . . . 15 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
86	79, 64, 85	mp2an 692	. . . . . . . . . . . . . 14 ⊢ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ (vol*‘∪ ran ((,) ∘ 𝑓))
87		xrletr 13200	. . . . . . . . . . . . . . . 16 ⊢ (((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ* ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ*) → (((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
88	82, 66, 87	mp3an12 1453	. . . . . . . . . . . . . . 15 ⊢ (((vol*‘𝐴) + 1) ∈ ℝ* → (((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
89	60, 88	syl 17	. . . . . . . . . . . . . 14 ⊢ ((vol*‘𝐴) ∈ ℝ → (((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
90	86, 89	mpani 696	. . . . . . . . . . . . 13 ⊢ ((vol*‘𝐴) ∈ ℝ → ((vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
91	90	ad4antlr 733	. . . . . . . . . . . 12 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ 𝐴 ⊆ ∪ ran ((,) ∘ 𝑓)) → ((vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
92	91	impr 454	. . . . . . . . . . 11 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1))
93		ovolge0 25516	. . . . . . . . . . . 12 ⊢ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ → 0 ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
94	80, 93	ax-mp 5	. . . . . . . . . . 11 ⊢ 0 ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))
95	92, 94	jctil 519	. . . . . . . . . 10 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (0 ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1)))
96		xrrege0 13216	. . . . . . . . . 10 ⊢ ((((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ) ∧ (0 ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
97	84, 95, 96	syl2anc 584	. . . . . . . . 9 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
98		resubcl 11573	. . . . . . . . . . . . . 14 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ 𝑢 ∈ ℝ) → ((vol*‘𝐴) − 𝑢) ∈ ℝ)
99	98	adantrr 717	. . . . . . . . . . . . 13 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ((vol*‘𝐴) − 𝑢) ∈ ℝ)
100		posdif 11756	. . . . . . . . . . . . . . . 16 ⊢ ((𝑢 ∈ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝑢 < (vol*‘𝐴) ↔ 0 < ((vol*‘𝐴) − 𝑢)))
101	100	ancoms 458	. . . . . . . . . . . . . . 15 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑢 < (vol*‘𝐴) ↔ 0 < ((vol*‘𝐴) − 𝑢)))
102	101	biimpd 229	. . . . . . . . . . . . . 14 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑢 < (vol*‘𝐴) → 0 < ((vol*‘𝐴) − 𝑢)))
103	102	impr 454	. . . . . . . . . . . . 13 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → 0 < ((vol*‘𝐴) − 𝑢))
104	99, 103	elrpd 13074	. . . . . . . . . . . 12 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ((vol*‘𝐴) − 𝑢) ∈ ℝ+)
105	104	rphalfcld 13089	. . . . . . . . . . 11 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ+)
106	3, 105	sylan 580	. . . . . . . . . 10 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ+)
107	106	adantr 480	. . . . . . . . 9 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ+)
108		eqid 2737	. . . . . . . . . . 11 ⊢ seq1( + , ((abs ∘ − ) ∘ 𝑔)) = seq1( + , ((abs ∘ − ) ∘ 𝑔))
109	108	ovolgelb 25515	. . . . . . . . . 10 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ℝ ∧ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ ∧ (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
110	80, 109	mp3an1 1450	. . . . . . . . 9 ⊢ (((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ ∧ (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ+) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
111	97, 107, 110	syl2anc 584	. . . . . . . 8 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
112		elmapi 8889	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ) → 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)))
113		ssid 4006	. . . . . . . . . . . . . 14 ⊢ ∪ ran ((,) ∘ 𝑔) ⊆ ∪ ran ((,) ∘ 𝑔)
114	108	ovollb 25514	. . . . . . . . . . . . . 14 ⊢ ((𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) ∧ ∪ ran ((,) ∘ 𝑔) ⊆ ∪ ran ((,) ∘ 𝑔)) → (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))
115	113, 114	mpan2 691	. . . . . . . . . . . . 13 ⊢ (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))
116	115	adantl 481	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ))
117		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ ((abs ∘ − ) ∘ 𝑔) = ((abs ∘ − ) ∘ 𝑔)
118	117, 108	ovolsf 25507	. . . . . . . . . . . . . 14 ⊢ (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → seq1( + , ((abs ∘ − ) ∘ 𝑔)):ℕ⟶(0[,)+∞))
119		frn 6743	. . . . . . . . . . . . . . 15 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑔)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑔)) ⊆ (0[,)+∞))
120	119, 55	sstrdi 3996	. . . . . . . . . . . . . 14 ⊢ (seq1( + , ((abs ∘ − ) ∘ 𝑔)):ℕ⟶(0[,)+∞) → ran seq1( + , ((abs ∘ − ) ∘ 𝑔)) ⊆ ℝ*)
121		supxrcl 13357	. . . . . . . . . . . . . 14 ⊢ (ran seq1( + , ((abs ∘ − ) ∘ 𝑔)) ⊆ ℝ* → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ*)
122	118, 120, 121	3syl 18	. . . . . . . . . . . . 13 ⊢ (𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ)) → sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ*)
123	99	rehalfcld 12513	. . . . . . . . . . . . . . . . 17 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ)
124	3, 123	sylan 580	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ)
125	124	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ)
126	97, 125	readdcld 11290	. . . . . . . . . . . . . 14 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ)
127	126	rexrd 11311	. . . . . . . . . . . . 13 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ*)
128		rncoss 5986	. . . . . . . . . . . . . . . . 17 ⊢ ran ((,) ∘ 𝑔) ⊆ ran (,)
129	128	unissi 4916	. . . . . . . . . . . . . . . 16 ⊢ ∪ ran ((,) ∘ 𝑔) ⊆ ∪ ran (,)
130	129, 63	sseqtrri 4033	. . . . . . . . . . . . . . 15 ⊢ ∪ ran ((,) ∘ 𝑔) ⊆ ℝ
131		ovolcl 25513	. . . . . . . . . . . . . . 15 ⊢ (∪ ran ((,) ∘ 𝑔) ⊆ ℝ → (vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℝ*)
132	130, 131	ax-mp 5	. . . . . . . . . . . . . 14 ⊢ (vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℝ*
133		xrletr 13200	. . . . . . . . . . . . . 14 ⊢ (((vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℝ* ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ* ∧ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ*) → (((vol*‘∪ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
134	132, 133	mp3an1 1450	. . . . . . . . . . . . 13 ⊢ ((sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ* ∧ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ*) → (((vol*‘∪ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
135	122, 127, 134	syl2anr 597	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (((vol*‘∪ ran ((,) ∘ 𝑔)) ≤ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
136	116, 135	mpand 695	. . . . . . . . . . 11 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ 𝑔:ℕ⟶( ≤ ∩ (ℝ × ℝ))) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) → (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
137	112, 136	sylan2 593	. . . . . . . . . 10 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → (sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) → (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
138	137	anim2d 612	. . . . . . . . 9 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ 𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)) → (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))))
139	138	reximdva 3168	. . . . . . . 8 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ − ) ∘ 𝑔)), ℝ*, < ) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))))
140	111, 139	mpd 15	. . . . . . 7 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
141		rexex 3076	. . . . . . 7 ⊢ (∃𝑔 ∈ (( ≤ ∩ (ℝ × ℝ)) ↑m ℕ)((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → ∃𝑔((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
142	140, 141	syl 17	. . . . . 6 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ∃𝑔((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
143	59, 66	jctil 519	. . . . . . . . . . . 12 ⊢ ((vol*‘𝐴) ∈ ℝ → ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ))
144	143	ad3antlr 731	. . . . . . . . . . 11 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ))
145		ovolge0 25516	. . . . . . . . . . . . . 14 ⊢ (∪ ran ((,) ∘ 𝑓) ⊆ ℝ → 0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)))
146	64, 145	ax-mp 5	. . . . . . . . . . . . 13 ⊢ 0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓))
147	146	jctl 523	. . . . . . . . . . . 12 ⊢ ((vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1) → (0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
148	147	adantl 481	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → (0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)))
149		xrrege0 13216	. . . . . . . . . . 11 ⊢ ((((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*‘𝐴) + 1) ∈ ℝ) ∧ (0 ≤ (vol*‘∪ ran ((,) ∘ 𝑓)) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
150	144, 148, 149	syl2an 596	. . . . . . . . . 10 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
151	150, 125	resubcld 11691	. . . . . . . . 9 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ)
152	150, 107	ltsubrpd 13109	. . . . . . . . 9 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘∪ ran ((,) ∘ 𝑓)))
153		retop 24782	. . . . . . . . . . 11 ⊢ (topGen‘ran (,)) ∈ Top
154		retopbas 24781	. . . . . . . . . . . . 13 ⊢ ran (,) ∈ TopBases
155		bastg 22973	. . . . . . . . . . . . 13 ⊢ (ran (,) ∈ TopBases → ran (,) ⊆ (topGen‘ran (,)))
156	154, 155	ax-mp 5	. . . . . . . . . . . 12 ⊢ ran (,) ⊆ (topGen‘ran (,))
157	61, 156	sstri 3993	. . . . . . . . . . 11 ⊢ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))
158		uniopn 22903	. . . . . . . . . . 11 ⊢ (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) ⊆ (topGen‘ran (,))) → ∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)))
159	153, 157, 158	mp2an 692	. . . . . . . . . 10 ⊢ ∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,))
160		mblfinlem2 37665	. . . . . . . . . 10 ⊢ ((∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))
161	159, 160	mp3an1 1450	. . . . . . . . 9 ⊢ ((((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘∪ ran ((,) ∘ 𝑓))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))
162	151, 152, 161	syl2anc 584	. . . . . . . 8 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))
163	162	adantr 480	. . . . . . 7 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ∃𝑠 ∈ (Clsd‘(topGen‘ran (,)))(𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))
164		indif2 4281	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑔))) = ((𝑠 ∩ ℝ) ∖ ∪ ran ((,) ∘ 𝑔))
165	22	cldss 23037	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → 𝑠 ⊆ ℝ)
166		dfss2 3969	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ⊆ ℝ ↔ (𝑠 ∩ ℝ) = 𝑠)
167	165, 166	sylib 218	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∩ ℝ) = 𝑠)
168	167	difeq1d 4125	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → ((𝑠 ∩ ℝ) ∖ ∪ ran ((,) ∘ 𝑔)) = (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))
169	164, 168	eqtrid 2789	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑔))) = (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))
170	128, 156	sstri 3993	. . . . . . . . . . . . . . . . 17 ⊢ ran ((,) ∘ 𝑔) ⊆ (topGen‘ran (,))
171		uniopn 22903	. . . . . . . . . . . . . . . . 17 ⊢ (((topGen‘ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑔) ⊆ (topGen‘ran (,))) → ∪ ran ((,) ∘ 𝑔) ∈ (topGen‘ran (,)))
172	153, 170, 171	mp2an 692	. . . . . . . . . . . . . . . 16 ⊢ ∪ ran ((,) ∘ 𝑔) ∈ (topGen‘ran (,))
173	22	opncld 23041	. . . . . . . . . . . . . . . 16 ⊢ (((topGen‘ran (,)) ∈ Top ∧ ∪ ran ((,) ∘ 𝑔) ∈ (topGen‘ran (,))) → (ℝ ∖ ∪ ran ((,) ∘ 𝑔)) ∈ (Clsd‘(topGen‘ran (,))))
174	153, 172, 173	mp2an 692	. . . . . . . . . . . . . . 15 ⊢ (ℝ ∖ ∪ ran ((,) ∘ 𝑔)) ∈ (Clsd‘(topGen‘ran (,)))
175		incld 23051	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (ℝ ∖ ∪ ran ((,) ∘ 𝑔)) ∈ (Clsd‘(topGen‘ran (,)))) → (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑔))) ∈ (Clsd‘(topGen‘ran (,))))
176	174, 175	mpan2 691	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∩ (ℝ ∖ ∪ ran ((,) ∘ 𝑔))) ∈ (Clsd‘(topGen‘ran (,))))
177	169, 176	eqeltrrd 2842	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ∈ (Clsd‘(topGen‘ran (,))))
178		simpr 484	. . . . . . . . . . . . . . 15 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑠 ⊆ ∪ ran ((,) ∘ 𝑓)) → 𝑠 ⊆ ∪ ran ((,) ∘ 𝑓))
179		simpl 482	. . . . . . . . . . . . . . 15 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑠 ⊆ ∪ ran ((,) ∘ 𝑓)) → (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔))
180	178, 179	ssdif2d 4148	. . . . . . . . . . . . . 14 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑠 ⊆ ∪ ran ((,) ∘ 𝑓)) → (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ⊆ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
181		dfin4 4278	. . . . . . . . . . . . . . 15 ⊢ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) = (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))
182		inss2 4238	. . . . . . . . . . . . . . 15 ⊢ (∪ ran ((,) ∘ 𝑓) ∩ 𝐴) ⊆ 𝐴
183	181, 182	eqsstrri 4031	. . . . . . . . . . . . . 14 ⊢ (∪ ran ((,) ∘ 𝑓) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ 𝐴
184	180, 183	sstrdi 3996	. . . . . . . . . . . . 13 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑠 ⊆ ∪ ran ((,) ∘ 𝑓)) → (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ⊆ 𝐴)
185		sseq1 4009	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) → (𝑏 ⊆ 𝐴 ↔ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ⊆ 𝐴))
186	185	anbi1d 631	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) → ((𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏)) ↔ ((𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏))))
187		fveq2 6906	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) = 𝑏 → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏))
188	187	eqcoms 2745	. . . . . . . . . . . . . . . 16 ⊢ (𝑏 = (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏))
189	188	biantrud 531	. . . . . . . . . . . . . . 15 ⊢ (𝑏 = (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) → ((𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ⊆ 𝐴 ↔ ((𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏))))
190	186, 189	bitr4d 282	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) → ((𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏)) ↔ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ⊆ 𝐴))
191	190	rspcev 3622	. . . . . . . . . . . . 13 ⊢ (((𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ⊆ 𝐴) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
192	177, 184, 191	syl2an 596	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑠 ⊆ ∪ ran ((,) ∘ 𝑓))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
193	192	an12s 649	. . . . . . . . . . 11 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ 𝑠 ⊆ ∪ ran ((,) ∘ 𝑓))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
194	193	adantrrr 725	. . . . . . . . . 10 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
195	194	adantlr 715	. . . . . . . . 9 ⊢ ((((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
196	195	adantll 714	. . . . . . . 8 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
197		difss 4136	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) ⊆ 𝐴
198		ovolsscl 25521	. . . . . . . . . . . 12 ⊢ (((𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) ∈ ℝ)
199	197, 198	mp3an1 1450	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) ∈ ℝ)
200	199	ad5antr 734	. . . . . . . . . 10 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) ∈ ℝ)
201		simp-6r 788	. . . . . . . . . 10 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘𝐴) ∈ ℝ)
202		simpl 482	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴)) → 𝑢 ∈ ℝ)
203	202	ad4antlr 733	. . . . . . . . . 10 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → 𝑢 ∈ ℝ)
204		difdif2 4296	. . . . . . . . . . . 12 ⊢ (𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = ((𝐴 ∖ 𝑠) ∪ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))
205	204	fveq2i 6909	. . . . . . . . . . 11 ⊢ (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) = (vol*‘((𝐴 ∖ 𝑠) ∪ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔))))
206		difss 4136	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∖ 𝑠) ⊆ 𝐴
207		inss1 4237	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)) ⊆ 𝐴
208	206, 207	unssi 4191	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∖ 𝑠) ∪ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔))) ⊆ 𝐴
209		ovolsscl 25521	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∖ 𝑠) ∪ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔))) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴 ∖ 𝑠) ∪ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))) ∈ ℝ)
210	208, 209	mp3an1 1450	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴 ∖ 𝑠) ∪ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))) ∈ ℝ)
211	210	ad5antr 734	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘((𝐴 ∖ 𝑠) ∪ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))) ∈ ℝ)
212		ovolsscl 25521	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∖ 𝑠) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∖ 𝑠)) ∈ ℝ)
213	206, 212	mp3an1 1450	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∖ 𝑠)) ∈ ℝ)
214	213	ad5antr 734	. . . . . . . . . . . . 13 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∖ 𝑠)) ∈ ℝ)
215		ovolsscl 25521	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∩ ∪ ran ((,) ∘ 𝑔)) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔))) ∈ ℝ)
216	207, 215	mp3an1 1450	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔))) ∈ ℝ)
217	216	ad5antr 734	. . . . . . . . . . . . 13 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔))) ∈ ℝ)
218	214, 217	readdcld 11290	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘(𝐴 ∖ 𝑠)) + (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))) ∈ ℝ)
219	3, 202, 98	syl2an 596	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ((vol*‘𝐴) − 𝑢) ∈ ℝ)
220	219	ad3antrrr 730	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘𝐴) − 𝑢) ∈ ℝ)
221		ssdifss 4140	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ℝ → (𝐴 ∖ 𝑠) ⊆ ℝ)
222	221	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴 ∖ 𝑠) ⊆ ℝ)
223		ssinss1 4246	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ℝ → (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)) ⊆ ℝ)
224	223	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)) ⊆ ℝ)
225		ovolun 25534	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∖ 𝑠) ⊆ ℝ ∧ (vol*‘(𝐴 ∖ 𝑠)) ∈ ℝ) ∧ ((𝐴 ∩ ∪ ran ((,) ∘ 𝑔)) ⊆ ℝ ∧ (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔))) ∈ ℝ)) → (vol*‘((𝐴 ∖ 𝑠) ∪ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))) ≤ ((vol*‘(𝐴 ∖ 𝑠)) + (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))))
226	222, 213, 224, 216, 225	syl22anc 839	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘((𝐴 ∖ 𝑠) ∪ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))) ≤ ((vol*‘(𝐴 ∖ 𝑠)) + (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))))
227	226	ad5antr 734	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘((𝐴 ∖ 𝑠) ∪ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))) ≤ ((vol*‘(𝐴 ∖ 𝑠)) + (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))))
228	124	ad2antrr 726	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ)
229	228	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (((vol*‘𝐴) − 𝑢) / 2) ∈ ℝ)
230	150	ad2antrr 726	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
231		simprl 771	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠))) → 𝑠 ⊆ ∪ ran ((,) ∘ 𝑓))
232	150	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ)
233		ovolsscl 25521	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑠) ∈ ℝ)
234	64, 233	mp3an2 1451	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘𝑠) ∈ ℝ)
235	231, 232, 234	syl2anr 597	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘𝑠) ∈ ℝ)
236	230, 235	resubcld 11691	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)) ∈ ℝ)
237		ssdif 4144	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) → (𝐴 ∖ 𝑠) ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝑠))
238		difss 4136	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∪ ran ((,) ∘ 𝑓) ∖ 𝑠) ⊆ ∪ ran ((,) ∘ 𝑓)
239	238, 64	sstri 3993	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ ran ((,) ∘ 𝑓) ∖ 𝑠) ⊆ ℝ
240		ovolss 25520	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐴 ∖ 𝑠) ⊆ (∪ ran ((,) ∘ 𝑓) ∖ 𝑠) ∧ (∪ ran ((,) ∘ 𝑓) ∖ 𝑠) ⊆ ℝ) → (vol*‘(𝐴 ∖ 𝑠)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)))
241	237, 239, 240	sylancl 586	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) → (vol*‘(𝐴 ∖ 𝑠)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)))
242	241	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1)) → (vol*‘(𝐴 ∖ 𝑠)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)))
243	242	ad3antlr 731	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∖ 𝑠)) ≤ (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)))
244		eleq1w 2824	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑏 = 𝑠 → (𝑏 ∈ dom vol ↔ 𝑠 ∈ dom vol))
245	244, 32	vtoclga 3577	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → 𝑠 ∈ dom vol)
246	245	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠))) → 𝑠 ∈ dom vol)
247		mblsplit 25567	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑠 ∈ dom vol ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘∪ ran ((,) ∘ 𝑓)) = ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠))))
248	64, 247	mp3an2 1451	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑠 ∈ dom vol ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘∪ ran ((,) ∘ 𝑓)) = ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠))))
249	246, 232, 248	syl2anr 597	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘∪ ran ((,) ∘ 𝑓)) = ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠))))
250	249	eqcomd 2743	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠))) = (vol*‘∪ ran ((,) ∘ 𝑓)))
251	230	recnd 11289	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℂ)
252		inss1 4237	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∪ ran ((,) ∘ 𝑓) ∩ 𝑠) ⊆ ∪ ran ((,) ∘ 𝑓)
253		ovolsscl 25521	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((∪ ran ((,) ∘ 𝑓) ∩ 𝑠) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
254	252, 64, 253	mp3an12 1453	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
255	150, 254	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
256	255	ad2antrr 726	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
257	256	recnd 11289	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℂ)
258		ovolsscl 25521	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝑠) ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ∪ ran ((,) ∘ 𝑓) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)) ∈ ℝ)
259	238, 64, 258	mp3an12 1453	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((vol*‘∪ ran ((,) ∘ 𝑓)) ∈ ℝ → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)) ∈ ℝ)
260	150, 259	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)) ∈ ℝ)
261	260	recnd 11289	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)) ∈ ℂ)
262	261	ad2antrr 726	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)) ∈ ℂ)
263	251, 257, 262	subaddd 11638	. . . . . . . . . . . . . . . . . 18 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠))) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)) ↔ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠))) = (vol*‘∪ ran ((,) ∘ 𝑓))))
264	250, 263	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠))) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)))
265		sseqin2 4223	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ↔ (∪ ran ((,) ∘ 𝑓) ∩ 𝑠) = 𝑠)
266	265	biimpi 216	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) → (∪ ran ((,) ∘ 𝑓) ∩ 𝑠) = 𝑠)
267	266	fveq2d 6910	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠)) = (vol*‘𝑠))
268	267	oveq2d 7447	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠))) = ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)))
269	268	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠))) = ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)))
270	269	ad2antll 729	. . . . . . . . . . . . . . . . 17 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∩ 𝑠))) = ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)))
271	264, 270	eqtr3d 2779	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝑠)) = ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)))
272	243, 271	breqtrd 5169	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∖ 𝑠)) ≤ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)))
273		simprrr 782	. . . . . . . . . . . . . . . 16 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠))
274	230, 229, 235, 273	ltsub23d 11868	. . . . . . . . . . . . . . 15 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘∪ ran ((,) ∘ 𝑓)) − (vol*‘𝑠)) < (((vol*‘𝐴) − 𝑢) / 2))
275	214, 236, 229, 272, 274	lelttrd 11419	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∖ 𝑠)) < (((vol*‘𝐴) − 𝑢) / 2))
276	216	ad4antr 732	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔))) ∈ ℝ)
277	126, 132	jctil 519	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ((vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℝ* ∧ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ))
278		simpr 484	. . . . . . . . . . . . . . . . . . 19 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))
279		ovolge0 25516	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∪ ran ((,) ∘ 𝑔) ⊆ ℝ → 0 ≤ (vol*‘∪ ran ((,) ∘ 𝑔)))
280	130, 279	ax-mp 5	. . . . . . . . . . . . . . . . . . 19 ⊢ 0 ≤ (vol*‘∪ ran ((,) ∘ 𝑔))
281	278, 280	jctil 519	. . . . . . . . . . . . . . . . . 18 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) → (0 ≤ (vol*‘∪ ran ((,) ∘ 𝑔)) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
282		xrrege0 13216	. . . . . . . . . . . . . . . . . 18 ⊢ ((((vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℝ* ∧ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)) ∈ ℝ) ∧ (0 ≤ (vol*‘∪ ran ((,) ∘ 𝑔)) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℝ)
283	277, 281, 282	syl2an 596	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℝ)
284		difss 4136	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ ∪ ran ((,) ∘ 𝑔)
285		ovolsscl 25521	. . . . . . . . . . . . . . . . . 18 ⊢ (((∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ ∪ ran ((,) ∘ 𝑔) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℝ) → (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ∈ ℝ)
286	284, 130, 285	mp3an12 1453	. . . . . . . . . . . . . . . . 17 ⊢ ((vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℝ → (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ∈ ℝ)
287	283, 286	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ∈ ℝ)
288		ssun2 4179	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ ran ((,) ∘ 𝑔) ∩ 𝐴) ⊆ ((∪ ran ((,) ∘ 𝑔) ∖ ∪ ran ((,) ∘ 𝑓)) ∪ (∪ ran ((,) ∘ 𝑔) ∩ 𝐴))
289		incom 4209	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)) = (∪ ran ((,) ∘ 𝑔) ∩ 𝐴)
290		difdif2 4296	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) = ((∪ ran ((,) ∘ 𝑔) ∖ ∪ ran ((,) ∘ 𝑓)) ∪ (∪ ran ((,) ∘ 𝑔) ∩ 𝐴))
291	288, 289, 290	3sstr4i 4035	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)) ⊆ (∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))
292	284, 130	sstri 3993	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ ℝ
293	291, 292	pm3.2i 470	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐴 ∩ ∪ ran ((,) ∘ 𝑔)) ⊆ (∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ (∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ ℝ)
294		ovolss 25520	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐴 ∩ ∪ ran ((,) ∘ 𝑔)) ⊆ (∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∧ (∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ⊆ ℝ) → (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔))) ≤ (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))))
295	293, 294	mp1i 13	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔))) ≤ (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))))
296		opnmbl 25637	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∪ ran ((,) ∘ 𝑓) ∈ (topGen‘ran (,)) → ∪ ran ((,) ∘ 𝑓) ∈ dom vol)
297	159, 296	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ∪ ran ((,) ∘ 𝑓) ∈ dom vol
298		difmbl 25578	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∪ ran ((,) ∘ 𝑓) ∈ dom vol ∧ 𝐴 ∈ dom vol) → (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∈ dom vol)
299	297, 298	mpan 690	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐴 ∈ dom vol → (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∈ dom vol)
300	299	ad4antlr 733	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∈ dom vol)
301		mblsplit 25567	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∈ dom vol ∧ ∪ ran ((,) ∘ 𝑔) ⊆ ℝ ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℝ) → (vol*‘∪ ran ((,) ∘ 𝑔)) = ((vol*‘(∪ ran ((,) ∘ 𝑔) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))))
302	130, 301	mp3an2 1451	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ∈ dom vol ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℝ) → (vol*‘∪ ran ((,) ∘ 𝑔)) = ((vol*‘(∪ ran ((,) ∘ 𝑔) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))))
303	300, 283, 302	syl2anc 584	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘∪ ran ((,) ∘ 𝑔)) = ((vol*‘(∪ ran ((,) ∘ 𝑔) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))))
304		sseqin2 4223	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ↔ (∪ ran ((,) ∘ 𝑔) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) = (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))
305	304	biimpi 216	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) → (∪ ran ((,) ∘ 𝑔) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) = (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))
306	305	fveq2d 6910	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) → (vol*‘(∪ ran ((,) ∘ 𝑔) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) = (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))
307	306	oveq1d 7446	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) → ((vol*‘(∪ ran ((,) ∘ 𝑔) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))) = ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))))
308	307	ad2antrl 728	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ((vol*‘(∪ ran ((,) ∘ 𝑔) ∩ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) + (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))) = ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))))
309	303, 308	eqtr2d 2778	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))) = (vol*‘∪ ran ((,) ∘ 𝑔)))
310	283	recnd 11289	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘∪ ran ((,) ∘ 𝑔)) ∈ ℂ)
311	97	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℝ)
312	311	recnd 11289	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) ∈ ℂ)
313	287	recnd 11289	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ∈ ℂ)
314	310, 312, 313	subaddd 11638	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (((vol*‘∪ ran ((,) ∘ 𝑔)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) = (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ↔ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴)))) = (vol*‘∪ ran ((,) ∘ 𝑔))))
315	309, 314	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ((vol*‘∪ ran ((,) ∘ 𝑔)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) = (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))))
316		simprr 773	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))
317	283, 311, 228	lesubadd2d 11862	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (((vol*‘∪ ran ((,) ∘ 𝑔)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ≤ (((vol*‘𝐴) − 𝑢) / 2) ↔ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))))
318	316, 317	mpbird 257	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ((vol*‘∪ ran ((,) ∘ 𝑔)) − (vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ≤ (((vol*‘𝐴) − 𝑢) / 2))
319	315, 318	eqbrtrrd 5167	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘(∪ ran ((,) ∘ 𝑔) ∖ (∪ ran ((,) ∘ 𝑓) ∖ 𝐴))) ≤ (((vol*‘𝐴) − 𝑢) / 2))
320	276, 287, 228, 295, 319	letrd 11418	. . . . . . . . . . . . . . 15 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔))) ≤ (((vol*‘𝐴) − 𝑢) / 2))
321	320	adantr 480	. . . . . . . . . . . . . 14 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔))) ≤ (((vol*‘𝐴) − 𝑢) / 2))
322	214, 217, 229, 229, 275, 321	ltleaddd 11884	. . . . . . . . . . . . 13 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘(𝐴 ∖ 𝑠)) + (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))) < ((((vol*‘𝐴) − 𝑢) / 2) + (((vol*‘𝐴) − 𝑢) / 2)))
323	98	recnd 11289	. . . . . . . . . . . . . . . . 17 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ 𝑢 ∈ ℝ) → ((vol*‘𝐴) − 𝑢) ∈ ℂ)
324	323	2halvesd 12512	. . . . . . . . . . . . . . . 16 ⊢ (((vol*‘𝐴) ∈ ℝ ∧ 𝑢 ∈ ℝ) → ((((vol*‘𝐴) − 𝑢) / 2) + (((vol*‘𝐴) − 𝑢) / 2)) = ((vol*‘𝐴) − 𝑢))
325	324	adantll 714	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝑢 ∈ ℝ) → ((((vol*‘𝐴) − 𝑢) / 2) + (((vol*‘𝐴) − 𝑢) / 2)) = ((vol*‘𝐴) − 𝑢))
326	325	ad2ant2r 747	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ((((vol*‘𝐴) − 𝑢) / 2) + (((vol*‘𝐴) − 𝑢) / 2)) = ((vol*‘𝐴) − 𝑢))
327	326	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((((vol*‘𝐴) − 𝑢) / 2) + (((vol*‘𝐴) − 𝑢) / 2)) = ((vol*‘𝐴) − 𝑢))
328	322, 327	breqtrd 5169	. . . . . . . . . . . 12 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘(𝐴 ∖ 𝑠)) + (vol*‘(𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))) < ((vol*‘𝐴) − 𝑢))
329	211, 218, 220, 227, 328	lelttrd 11419	. . . . . . . . . . 11 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘((𝐴 ∖ 𝑠) ∪ (𝐴 ∩ ∪ ran ((,) ∘ 𝑔)))) < ((vol*‘𝐴) − 𝑢))
330	205, 329	eqbrtrid 5178	. . . . . . . . . 10 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) < ((vol*‘𝐴) − 𝑢))
331	200, 201, 203, 330	ltsub13d 11869	. . . . . . . . 9 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → 𝑢 < ((vol*‘𝐴) − (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))))
332		opnmbl 25637	. . . . . . . . . . . . . . . 16 ⊢ (∪ ran ((,) ∘ 𝑔) ∈ (topGen‘ran (,)) → ∪ ran ((,) ∘ 𝑔) ∈ dom vol)
333	172, 332	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ∪ ran ((,) ∘ 𝑔) ∈ dom vol
334		difmbl 25578	. . . . . . . . . . . . . . 15 ⊢ ((𝑠 ∈ dom vol ∧ ∪ ran ((,) ∘ 𝑔) ∈ dom vol) → (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ∈ dom vol)
335	245, 333, 334	sylancl 586	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ∈ dom vol)
336		mblvol 25565	. . . . . . . . . . . . . 14 ⊢ ((𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ∈ dom vol → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol*‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))
337	335, 336	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol*‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))
338	337	ad2antrl 728	. . . . . . . . . . . 12 ⊢ ((((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol*‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))
339		sseqin2 4223	. . . . . . . . . . . . . . . 16 ⊢ ((𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ⊆ 𝐴 ↔ (𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))
340	184, 339	sylib 218	. . . . . . . . . . . . . . 15 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑠 ⊆ ∪ ran ((,) ∘ 𝑓)) → (𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))
341	340	fveq2d 6910	. . . . . . . . . . . . . 14 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ 𝑠 ⊆ ∪ ran ((,) ∘ 𝑓)) → (vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) = (vol*‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))
342	341	adantrr 717	. . . . . . . . . . . . 13 ⊢ (((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠))) → (vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) = (vol*‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))
343	342	ad2ant2rl 749	. . . . . . . . . . . 12 ⊢ ((((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) = (vol*‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))
344	338, 343	eqtr4d 2780	. . . . . . . . . . 11 ⊢ ((((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))))
345	344	adantll 714	. . . . . . . . . 10 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))))
346	335	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠))) → (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ∈ dom vol)
347		simp-4l 783	. . . . . . . . . . . 12 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ))
348		mblsplit 25567	. . . . . . . . . . . . . 14 ⊢ (((𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ∈ dom vol ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘𝐴) = ((vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))))
349	348	3expb 1121	. . . . . . . . . . . . 13 ⊢ (((𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ∈ dom vol ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) → (vol*‘𝐴) = ((vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))))
350	349	eqcomd 2743	. . . . . . . . . . . 12 ⊢ (((𝑠 ∖ ∪ ran ((,) ∘ 𝑔)) ∈ dom vol ∧ (𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ)) → ((vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))) = (vol*‘𝐴))
351	346, 347, 350	syl2anr 597	. . . . . . . . . . 11 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))) = (vol*‘𝐴))
352		recn 11245	. . . . . . . . . . . . . 14 ⊢ ((vol*‘𝐴) ∈ ℝ → (vol*‘𝐴) ∈ ℂ)
353	352	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘𝐴) ∈ ℂ)
354	199	recnd 11289	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) ∈ ℂ)
355		inss1 4237	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) ⊆ 𝐴
356		ovolsscl 25521	. . . . . . . . . . . . . . 15 ⊢ (((𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) ⊆ 𝐴 ∧ 𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) ∈ ℝ)
357	355, 356	mp3an1 1450	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) ∈ ℝ)
358	357	recnd 11289	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) ∈ ℂ)
359	353, 354, 358	subadd2d 11639	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) → (((vol*‘𝐴) − (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))) = (vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) ↔ ((vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))) = (vol*‘𝐴)))
360	359	ad5antr 734	. . . . . . . . . . 11 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (((vol*‘𝐴) − (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))) = (vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) ↔ ((vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) + (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))) = (vol*‘𝐴)))
361	351, 360	mpbird 257	. . . . . . . . . 10 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ((vol*‘𝐴) − (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))) = (vol*‘(𝐴 ∩ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))))
362	345, 361	eqtr4d 2780	. . . . . . . . 9 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = ((vol*‘𝐴) − (vol*‘(𝐴 ∖ (𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))))
363	331, 362	breqtrrd 5171	. . . . . . . 8 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → 𝑢 < (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))
364		fvex 6919	. . . . . . . . 9 ⊢ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) ∈ V
365		eqeq1 2741	. . . . . . . . . . . 12 ⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) → (𝑣 = (vol‘𝑏) ↔ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏)))
366	365	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) → ((𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏))))
367	366	rexbidv 3179	. . . . . . . . . 10 ⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏))))
368		breq2 5147	. . . . . . . . . 10 ⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) → (𝑢 < 𝑣 ↔ 𝑢 < (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))))
369	367, 368	anbi12d 632	. . . . . . . . 9 ⊢ (𝑣 = (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) → ((∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣) ↔ (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏)) ∧ 𝑢 < (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))))))
370	364, 369	spcev 3606	. . . . . . . 8 ⊢ ((∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔))) = (vol‘𝑏)) ∧ 𝑢 < (vol‘(𝑠 ∖ ∪ ran ((,) ∘ 𝑔)))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
371	196, 363, 370	syl2anc 584	. . . . . . 7 ⊢ (((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) ∧ (𝑠 ∈ (Clsd‘(topGen‘ran (,))) ∧ (𝑠 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ ((vol*‘∪ ran ((,) ∘ 𝑓)) − (((vol*‘𝐴) − 𝑢) / 2)) < (vol*‘𝑠)))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
372	163, 371	rexlimddv 3161	. . . . . 6 ⊢ ((((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) ∧ ((∪ ran ((,) ∘ 𝑓) ∖ 𝐴) ⊆ ∪ ran ((,) ∘ 𝑔) ∧ (vol*‘∪ ran ((,) ∘ 𝑔)) ≤ ((vol*‘(∪ ran ((,) ∘ 𝑓) ∖ 𝐴)) + (((vol*‘𝐴) − 𝑢) / 2)))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
373	142, 372	exlimddv 1935	. . . . 5 ⊢ (((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) ∧ (𝐴 ⊆ ∪ ran ((,) ∘ 𝑓) ∧ (vol*‘∪ ran ((,) ∘ 𝑓)) ≤ ((vol*‘𝐴) + 1))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
374	78, 373	exlimddv 1935	. . . 4 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
375		eqeq1 2741	. . . . . . 7 ⊢ (𝑦 = 𝑣 → (𝑦 = (vol‘𝑏) ↔ 𝑣 = (vol‘𝑏)))
376	375	anbi2d 630	. . . . . 6 ⊢ (𝑦 = 𝑣 → ((𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ (𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏))))
377	376	rexbidv 3179	. . . . 5 ⊢ (𝑦 = 𝑣 → (∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏)) ↔ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏))))
378	377	rexab 3700	. . . 4 ⊢ (∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣 ↔ ∃𝑣(∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑣 = (vol‘𝑏)) ∧ 𝑢 < 𝑣))
379	374, 378	sylibr 234	. . 3 ⊢ ((((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑢 ∈ ℝ ∧ 𝑢 < (vol*‘𝐴))) → ∃𝑣 ∈ {𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}𝑢 < 𝑣)
380	2, 3, 42, 379	eqsupd 9497	. 2 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ) = (vol*‘𝐴))
381	380	eqcomd 2743	1 ⊢ (((𝐴 ⊆ ℝ ∧ (vol*‘𝐴) ∈ ℝ) ∧ 𝐴 ∈ dom vol) → (vol*‘𝐴) = sup({𝑦 ∣ ∃𝑏 ∈ (Clsd‘(topGen‘ran (,)))(𝑏 ⊆ 𝐴 ∧ 𝑦 = (vol‘𝑏))}, ℝ, < ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∃wrex 3070   ∖ cdif 3948   ∪ cun 3949   ∩ cin 3950   ⊆ wss 3951  ∪ cuni 4907   class class class wbr 5143   Or wor 5591   × cxp 5683  dom cdm 5685  ran crn 5686   ∘ ccom 5689  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ↑_m cmap 8866  supcsup 9480  ℂcc 11153  ℝcr 11154  0cc0 11155  1c1 11156   + caddc 11158  +∞cpnf 11292  ℝ^*cxr 11294   < clt 11295   ≤ cle 11296   − cmin 11492   / cdiv 11920  ℕcn 12266  2c2 12321  ℝ⁺crp 13034  (,)cioo 13387  [,)cico 13389  seqcseq 14042  abscabs 15273  topGenctg 17482  Topctop 22899  TopBasesctb 22952  Clsdccld 23024  vol*covol 25497  volcvol 25498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-inf2 9681  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-disj 5111  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-pm 8869  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-fi 9451  df-sup 9482  df-inf 9483  df-oi 9550  df-dju 9941  df-card 9979  df-acn 9982  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-n0 12527  df-z 12614  df-uz 12879  df-q 12991  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-ioo 13391  df-ico 13393  df-icc 13394  df-fz 13548  df-fzo 13695  df-fl 13832  df-seq 14043  df-exp 14103  df-hash 14370  df-cj 15138  df-re 15139  df-im 15140  df-sqrt 15274  df-abs 15275  df-clim 15524  df-rlim 15525  df-sum 15723  df-rest 17467  df-topgen 17488  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-mopn 21360  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-cmp 23395  df-conn 23420  df-ovol 25499  df-vol 25500

This theorem is referenced by:  ismblfin  37668